Revise geometry with different types of rectangles and lines.
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@ -82,7 +82,7 @@ A collection of points satisfying \eqref{sec:partition-eq1} is called a
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\begin{definition}
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\lean{Common/Set/Intervals/Partition}{Set.Intervals.Partition}
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\lean{Common/Set/Partition}{Set.Partition}
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\end{definition}
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@ -114,7 +114,7 @@ That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$
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\begin{definition}
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\lean{Common/Set/Intervals/StepFunction}{Set.Intervals.StepFunction}
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\lean{Common/Geometry/StepFunction}{Geometry.StepFunction}
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\end{definition}
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@ -570,7 +570,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
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\begin{axiom}
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\leanPretty{Common/Real/Geometry/Area}{Nonnegative-Property}
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\leanPretty{Common/Geometry/Area}{Nonnegative-Property}
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{Nonnegative Property}
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\end{axiom}
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@ -583,7 +583,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
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\begin{axiom}
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\leanPretty{Common/Real/Geometry/Area}{Additive-Property}
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\leanPretty{Common/Geometry/Area}{Additive-Property}
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{Additive Property}
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\end{axiom}
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@ -596,7 +596,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
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\begin{axiom}
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\leanPretty{Common/Real/Geometry/Area}{Difference-Property}
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\leanPretty{Common/Geometry/Area}{Difference-Property}
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{Difference Property}
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\end{axiom}
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@ -609,7 +609,7 @@ If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
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\begin{axiom}
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\leanPretty{Common/Real/Geometry/Area}{Invariance-Under-Congruence}
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\leanPretty{Common/Geometry/Area}{Invariance-Under-Congruence}
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{Invariance Under Congruence}
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\end{axiom}
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@ -622,7 +622,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
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\begin{axiom}
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\leanPretty{Common/Real/Geometry/Area}{Choice-of-Scale}
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\leanPretty{Common/Geometry/Area}{Choice-of-Scale}
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{Choice of Scale}
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\end{axiom}
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@ -642,7 +642,7 @@ If there is one and only one number $c$ which satisfies the inequalities
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\begin{axiom}
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\leanPretty{Common/Real/Geometry/Area}{Exhaustion-Property}
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\leanPretty{Common/Geometry/Area}{Exhaustion-Property}
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{Exhaustion Property}
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\end{axiom}
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@ -1,9 +1,8 @@
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import Mathlib.Data.Real.Basic
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import Mathlib.Tactic.LibrarySearch
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import Common.Real.Floor
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import Common.Geometry.StepFunction
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import Common.Set.Basic
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import Common.Set.Intervals.StepFunction
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/-! # Apostol.Chapter_1_11 -/
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@ -133,8 +132,6 @@ theorem exercise_7b (ha : a > 0) (hb : b > 0) (hp : Nat.coprime a b)
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section
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open Set.Intervals
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/-- ### Exercise 8
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Let `S` be a set of points on the real line. The *characteristic function* of
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@ -1,7 +1,8 @@
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import Common.Real.Geometry.Rectangle
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import Common.Set.Intervals.StepFunction
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import Common.Geometry.Basic
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import Common.Geometry.Rectangle.Skew
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import Common.Geometry.StepFunction
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/-! # Common.Real.Geometry.Area
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/-! # Common.Geometry.Area
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An axiomatic foundation for the concept of *area*. These axioms are those
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outlined in [^1].
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@ -10,7 +11,7 @@ outlined in [^1].
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Introduction to Linear Algebra. 2nd ed. Vol. 1. 2 vols. Wiley, 1991.
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-/
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namespace Real.Geometry.Area
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namespace Geometry.Area
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/--
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All *measurable sets*, i.e. sets in the plane to which an area can be assigned.
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@ -18,7 +19,7 @@ All *measurable sets*, i.e. sets in the plane to which an area can be assigned.
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The existence of such a class is assumed in the axiomatic definition of area
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introduced by Apostol. He denotes this set of sets as `𝓜`.
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-/
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axiom 𝓜 : Set (Set ℝ²)
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axiom 𝓜 : Set (Set Point)
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/--
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A set function mapping every *measurable set* to a value denoting its area.
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@ -33,7 +34,7 @@ axiom area : ∀ ⦃x⦄, x ∈ 𝓜 → ℝ
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For each set `S` in `𝓜`, we have `a(S) ≥ 0`.
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-/
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axiom area_ge_zero {S : Set ℝ²} (h : S ∈ 𝓜): area h ≥ 0
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axiom area_ge_zero {S : Set Point} (h : S ∈ 𝓜): area h ≥ 0
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/-! ## Additive Property
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@ -41,14 +42,14 @@ If `S` and `T` are in `𝓜`, then `S ∪ T` in `𝓜`, `S ∩ T` in `𝓜`, and
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`a(S ∪ T) = a(S) + a(T) - a(S ∩ T)`.
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-/
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axiom measureable_imp_union_measurable {S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜)
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: S ∪ T ∈ 𝓜
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axiom measureable_imp_union_measurable {S T : Set Point}
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(hS : S ∈ 𝓜) (hT : T ∈ 𝓜) : S ∪ T ∈ 𝓜
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axiom measurable_imp_inter_measurable {S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜)
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: S ∩ T ∈ 𝓜
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axiom measurable_imp_inter_measurable {S T : Set Point}
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(hS : S ∈ 𝓜) (hT : T ∈ 𝓜) : S ∩ T ∈ 𝓜
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axiom union_area_eq_area_add_area_sub_inter_area
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜)
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{S T : Set Point} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜)
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: area (measureable_imp_union_measurable hS hT) =
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area hS + area hT - area (measurable_imp_inter_measurable hS hT)
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@ -59,11 +60,11 @@ If `S` and `T` are in `𝓜` with `S ⊆ T`, then `T - S` is in `𝓜` and
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-/
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axiom measureable_imp_diff_measurable
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜) (h : S ⊆ T)
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{S T : Set Point} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜) (h : S ⊆ T)
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: T \ S ∈ 𝓜
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axiom diff_area_eq_area_sub_area
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜) (h : S ⊆ T)
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{S T : Set Point} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜) (h : S ⊆ T)
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: area (measureable_imp_diff_measurable hS hT h) = area hT - area hS
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/-! ## Invariance Under Congruence
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@ -73,14 +74,13 @@ If a set `S` is in `𝓜` and if a set `T` is congruent to `S`, then `T` is also
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-/
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axiom measurable_congruent_imp_measurable
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (h : congruent S T)
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{S T : Set Point} (hS : S ∈ 𝓜) (h : congruent S T)
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: T ∈ 𝓜
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axiom congruent_imp_area_eq_area
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (h : congruent S T)
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{S T : Set Point} (hS : S ∈ 𝓜) (h : congruent S T)
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: area hS = area (measurable_congruent_imp_measurable hS h)
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/-! ## Choice of Scale
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(i) Every rectangle `R` is in `𝓜`.
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@ -88,10 +88,10 @@ axiom congruent_imp_area_eq_area
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(ii) If the edges of rectangle `R` have lengths `h` and `k`, then `a(R) = hk`.
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-/
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axiom rectangle_measurable (R : Rectangle)
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axiom rectangle_measurable (R : Rectangle.Skew)
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: R.toSet ∈ 𝓜
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axiom rectangle_area_eq_mul_edge_lengths (R : Rectangle)
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axiom rectangle_area_eq_mul_edge_lengths (R : Rectangle.Skew)
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: area (rectangle_measurable R) = R.width * R.height
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/-! ## Exhaustion Property
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@ -107,24 +107,24 @@ Every step region is measurable. This follows from the choice of scale axiom,
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and the fact all step regions are equivalent to the union of a collection of
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rectangles.
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-/
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theorem step_function_measurable (S : Set.Intervals.StepFunction ℝ)
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theorem step_function_measurable (S : StepFunction)
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: S.toSet ∈ 𝓜 := by
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sorry
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def forallSubsetsBetween (k : ℝ) (Q : Set ℝ²) :=
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∀ S T : Set.Intervals.StepFunction ℝ,
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def forallSubsetsBetween (k : ℝ) (Q : Set Point) :=
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∀ S T : StepFunction,
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(hS : S.toSet ⊆ Q) →
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(hT : Q ⊆ T.toSet) →
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area (step_function_measurable S) ≤ k ∧ k ≤ area (step_function_measurable T)
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axiom exhaustion_exists_unique_imp_measurable (Q : Set ℝ²)
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axiom exhaustion_exists_unique_imp_measurable (Q : Set Point)
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: (∃! k : ℝ, forallSubsetsBetween k Q)
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→ Q ∈ 𝓜
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axiom exhaustion_exists_unique_imp_area_eq (Q : Set ℝ²)
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axiom exhaustion_exists_unique_imp_area_eq (Q : Set Point)
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: ∃ k : ℝ,
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(h : forallSubsetsBetween k Q ∧
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(∀ x : ℝ, forallSubsetsBetween x Q → x = k))
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→ area (exhaustion_exists_unique_imp_measurable Q ⟨k, h⟩) = k
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end Real.Geometry.Area
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end Geometry.Area
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@ -0,0 +1,38 @@
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import Common.Geometry.Point
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/-! # Common.Geometry.Basic
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Additional theorems and definitions useful in the context of geometry.
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-/
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namespace Geometry
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/--
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Two sets `S` and `T` are `similar` **iff** there exists a one-to-one
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correspondence between `S` and `T` such that the distance between any two points
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`P, Q ∈ S` and corresponding points `P', Q' ∈ T` differ by some constant `α`. In
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other words, `α|PQ| = |P'Q'|`.
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-/
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def similar (S T : Set Point) : Prop :=
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∃ f : Point → Point, Function.Bijective f ∧
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∃ s : ℝ, ∀ x y : Point, x ∈ S ∧ y ∈ T →
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s * Point.dist x y = Point.dist (f x) (f y)
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/--
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Two sets are congruent if they are similar with a scaling factor of `1`.
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-/
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def congruent (S T : Set Point) : Prop :=
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∃ f : Point → Point, Function.Bijective f ∧
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∀ x y : Point, x ∈ S ∧ y ∈ T →
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Point.dist x y = Point.dist (f x) (f y)
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/--
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Any two `congruent` sets must be similar to one another.
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-/
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theorem congruent_similar {S T : Set Point} : congruent S T → similar S T := by
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intro hc
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let ⟨f, ⟨hf, hs⟩⟩ := hc
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conv at hs => intro x y hxy; arg 1; rw [← one_mul (Point.dist x y)]
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exact ⟨f, ⟨hf, ⟨1, hs⟩⟩⟩
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end Geometry
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@ -0,0 +1,24 @@
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import Common.Geometry.Point
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/-! # Common.Geometry.Line
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A representation of a line in the two-dimensional Cartesian plane.
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-/
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namespace Geometry
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/--
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A `Line` is represented in vector form (not to be confused with `Vector`).
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It is completely characterized by a point `P` on the line in question and a
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direction vector `dir` parallel to the line. Such a line is represented by
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equation
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```
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\vec{r} = \vec{OP} + t\vec{d},
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```
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where `O` denotes the origin and `t ∈ ℝ`.
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-/
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structure Line where
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p : Point
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dir : ℝ × ℝ
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end Geometry
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@ -0,0 +1,77 @@
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Real.Sqrt
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import Common.Real.Trigonometry
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/-! # Common.Geometry.Point
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A representation of a point in the two-dimensional Cartesian plane.
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-/
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namespace Geometry
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structure Point where
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x : ℝ
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y : ℝ
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noncomputable instance : DecidableEq Point := by
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intro p₁ p₂
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have ⟨x₁, y₁⟩ := p₁
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have ⟨x₂, y₂⟩ := p₂
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if hp : x₁ = x₂ ∧ y₁ = y₂ then
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rw [hp.left, hp.right]
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exact isTrue rfl
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else
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rw [not_and_or] at hp
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refine isFalse ?_
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intro h
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injection h with hx hy
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apply Or.elim hp
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· intro nx
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exact absurd hx nx
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· intro ny
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exact absurd hy ny
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namespace Point
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/--
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The function mapping a `Point` to a `2`-tuple of reals.
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-/
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def toTuple (p : Point) : ℝ × ℝ := (p.x, p.y)
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/--
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The function mapping a `2`-tuple of reals to a `Point`.
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-/
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def fromTuple (p : ℝ × ℝ) : Point := Point.mk p.1 p.2
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/--
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An isomorphism between a `Point` and a `2`-tuple.
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-/
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def isoTuple : Point ≃ ℝ × ℝ :=
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{
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toFun := toTuple,
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invFun := fromTuple,
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left_inv := by
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unfold Function.LeftInverse fromTuple toTuple
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simp,
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right_inv := by
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unfold Function.RightInverse Function.LeftInverse fromTuple toTuple
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simp
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}
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/--
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The length of the straight line segment connecting points `p` and `q`.
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-/
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noncomputable def dist (p q : Point) :=
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Real.sqrt ((abs (q.x - p.x)) ^ 2 + (abs (q.y - p.y)) ^ 2)
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/--
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The undirected angle at `p₂` between the line segments to `p₁` and `p₃`. If
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either of those points equals `p₂`, this is `π / 2`.
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-/
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noncomputable def angle (p₁ p₂ p₃ : Point) : ℝ :=
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Real.euclideanAngle (toTuple p₁) (toTuple p₂) (toTuple p₃)
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end Point
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end Geometry
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@ -0,0 +1,2 @@
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import Common.Geometry.Rectangle.Orthogonal
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import Common.Geometry.Rectangle.Skew
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@ -0,0 +1,182 @@
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import Mathlib.Data.Fin.Basic
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import Mathlib.Tactic.LibrarySearch
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import Common.Geometry.Point
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import Common.Geometry.Segment
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import Common.Geometry.Rectangle.Skew
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/-! # Common.Geometry.Rectangle.Orthogonal
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A characterization of an orthogonal rectangle.
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-/
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namespace Geometry.Rectangle
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/--
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An `Orthogonal` rectangle is characterized by two points on opposite corners. It
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is assumed the edges of the rectangle are parallel to the coordinate axes.
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A `Point` can alternatively be viewed as an `Orthogonal` rectangle in which the
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two points coincide. A horizontal or vertical `Segment` can alternatively be
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viewed as an `Orthogonal` rectangle with width or height (but not both) `0`.
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-/
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structure Orthogonal where
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bl : Point -- bottom left
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tr : Point -- top right
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namespace Orthogonal
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/--
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The width of the `Orthogonal` rectangle.
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-/
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def width (r : Orthogonal) := r.tr.x - r.bl.x
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/--
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The height of the `Orthogonal` rectangle.
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-/
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def height (r : Orthogonal) := r.tr.y - r.bl.y
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/--
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The top-left corner of the `Orthogonal` rectangle.
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-/
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def tl (r : Orthogonal) : Point := ⟨r.bl.x, r.bl.y + r.height⟩
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/--
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The bottom-right corner of the `Orthogonal` rectangle.
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-/
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def br (r : Orthogonal) : Point := ⟨r.bl.x + r.width, r.bl.y⟩
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/--
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An `Orthogonal` rectangle's top side is equal in length to its bottom side.
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-/
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theorem dist_top_eq_dist_bottom (r : Orthogonal)
|
||||
: Point.dist r.tl r.tr = Point.dist r.bl r.br := by
|
||||
unfold tl br Point.dist width height
|
||||
norm_num
|
||||
|
||||
/--
|
||||
An `Orthogonal` rectangle's left side is equal in length to its right side.
|
||||
-/
|
||||
theorem dist_left_eq_dist_right (r : Orthogonal)
|
||||
: Point.dist r.tl r.bl = Point.dist r.tr r.br := by
|
||||
unfold tl br Point.dist width height
|
||||
norm_num
|
||||
|
||||
/--
|
||||
Convert an `Orthogonal` rectangle into a `Skew` one.
|
||||
-/
|
||||
def toSkew (r : Orthogonal) : Skew := ⟨r.tl, r.bl, r.br, sorry⟩
|
||||
|
||||
/--
|
||||
The set of `Orthogonal` rectangles are embedded in the set of `Skew` rectangles.
|
||||
-/
|
||||
def skewEmbedding : Orthogonal ↪ Skew :=
|
||||
have : Function.Injective toSkew := by
|
||||
unfold Function.Injective
|
||||
intro r₁ r₂ h
|
||||
unfold toSkew at h
|
||||
have ⟨⟨blx₁, bly₁⟩, ⟨trx₁, try₁⟩⟩ := r₁
|
||||
have ⟨⟨blx₂, bry₂⟩, ⟨trx₂, try₂⟩⟩ := r₂
|
||||
simp
|
||||
simp at h
|
||||
unfold tl br width height at h
|
||||
simp at h
|
||||
exact ⟨⟨h.left.left, h.right.left.right⟩, ⟨h.right.right.left, h.left.right⟩⟩
|
||||
⟨toSkew, this⟩
|
||||
|
||||
/-! ## Point -/
|
||||
|
||||
/--
|
||||
A `Point` is an `Orthogonal` rectangle in which all points coincide.
|
||||
-/
|
||||
abbrev AsPoint := Subtype (fun r : Orthogonal => r.bl = r.tr)
|
||||
|
||||
namespace AsPoint
|
||||
|
||||
/--
|
||||
The function mapping an `Orthogonal` rectangle with all points coinciding to a
|
||||
`Point`.
|
||||
-/
|
||||
def toPoint (p : AsPoint) : Point := p.val.tl
|
||||
|
||||
/--
|
||||
The function mapping a `Point` to an `Orthogonal` rectangle with all points
|
||||
coinciding.
|
||||
-/
|
||||
def fromPoint (p : Point) : AsPoint := ⟨Orthogonal.mk p p, by simp⟩
|
||||
|
||||
/--
|
||||
An isomorphism between an `Orthogonal` rectangle with all points coinciding and
|
||||
a `Point`.
|
||||
-/
|
||||
def isoPoint : AsPoint ≃ Point :=
|
||||
{
|
||||
toFun := toPoint,
|
||||
invFun := fromPoint,
|
||||
left_inv := by
|
||||
unfold Function.LeftInverse fromPoint toPoint
|
||||
intro ⟨r, hr⟩
|
||||
congr
|
||||
repeat {
|
||||
simp only
|
||||
unfold tl height
|
||||
rw [hr]
|
||||
simp
|
||||
}
|
||||
right_inv := by
|
||||
unfold Function.RightInverse Function.LeftInverse fromPoint toPoint
|
||||
intro ⟨r, hr⟩
|
||||
unfold tl height
|
||||
simp
|
||||
}
|
||||
|
||||
/--
|
||||
The width of an `AsPoint` is `0`.
|
||||
-/
|
||||
theorem width_eq_zero (p : AsPoint) : p.val.width = 0 := by
|
||||
unfold Orthogonal.width
|
||||
rw [p.property]
|
||||
simp
|
||||
|
||||
/--
|
||||
The height of an `AsPoint` is `0`.
|
||||
-/
|
||||
theorem height_eq_zero (p : AsPoint) : p.val.height = 0 := by
|
||||
unfold Orthogonal.height
|
||||
rw [p.property]
|
||||
simp
|
||||
|
||||
end AsPoint
|
||||
|
||||
/-! ## Segment -/
|
||||
|
||||
/--
|
||||
A `Segment` is an `Orthogonal` rectangle either width or height equal to `0`.
|
||||
-/
|
||||
abbrev AsSegment := Subtype (fun r : Orthogonal =>
|
||||
(r.bl.x = r.tr.x ∧ r.bl.y ≠ r.tr.y) ∨ (r.bl.x ≠ r.tr.x ∧ r.bl.y = r.tr.y))
|
||||
|
||||
namespace AsSegment
|
||||
|
||||
/--
|
||||
Either the width or height of an `AsSegment` is zero.
|
||||
-/
|
||||
theorem width_or_height_eq_zero (s : AsSegment)
|
||||
: s.val.width = 0 ∨ s.val.height = 0 := by
|
||||
apply Or.elim s.property
|
||||
· intro h
|
||||
refine Or.inl ?_
|
||||
unfold width
|
||||
rw [h.left]
|
||||
simp
|
||||
· intro h
|
||||
refine Or.inr ?_
|
||||
unfold height
|
||||
rw [h.right]
|
||||
simp
|
||||
|
||||
end AsSegment
|
||||
|
||||
end Orthogonal
|
||||
|
||||
end Geometry.Rectangle
|
||||
|
|
@ -0,0 +1,176 @@
|
|||
import Common.Geometry.Point
|
||||
import Common.Geometry.Segment
|
||||
import Common.Real.Trigonometry
|
||||
|
||||
/-! # Common.Geometry.Rectangle.Skew
|
||||
|
||||
A characterization of a skew rectangle.
|
||||
-/
|
||||
|
||||
namespace Geometry.Rectangle
|
||||
|
||||
/--
|
||||
A `Skew` rectangle is characterized by three points and the angle formed between
|
||||
them.
|
||||
|
||||
A `Point` can alternatively be viewed as a `Skew` rectangle in which all three
|
||||
points coincide. A `Segment` can alternatively be viewed as a `Skew` rectangle
|
||||
in which two of the three points coincide. Note that, by definition of
|
||||
`Point.angle`, such a situation does indeed yield an angle of `π / 2`.
|
||||
-/
|
||||
structure Skew where
|
||||
tl : Point -- top left
|
||||
bl : Point -- bottom left
|
||||
br : Point -- bottom right
|
||||
has_right_angle : Point.angle tl bl br = Real.pi / 2
|
||||
|
||||
namespace Skew
|
||||
|
||||
/--
|
||||
The top-right corner of the `Skew` rectangle.
|
||||
-/
|
||||
def tr (r : Skew) : Point :=
|
||||
⟨r.tl.x + r.br.x - r.bl.x, r.tl.y + r.br.y - r.bl.y⟩
|
||||
|
||||
/--
|
||||
A `Skew` rectangle's top side is equal in length to its bottom side.
|
||||
-/
|
||||
theorem dist_top_eq_dist_bottom (r : Skew)
|
||||
: Point.dist r.tl r.tr = Point.dist r.bl r.br := by
|
||||
unfold tr Point.dist
|
||||
simp only
|
||||
repeat rw [add_comm, sub_right_comm, add_sub_cancel']
|
||||
|
||||
/--
|
||||
A `Skew` rectangle's left side is equal in length to its right side.
|
||||
-/
|
||||
theorem dist_left_eq_dist_right (r : Skew)
|
||||
: Point.dist r.tl r.bl = Point.dist r.tr r.br := by
|
||||
unfold tr Point.dist
|
||||
simp only
|
||||
repeat rw [
|
||||
sub_sub_eq_add_sub,
|
||||
add_comm,
|
||||
sub_add_eq_sub_sub,
|
||||
sub_right_comm,
|
||||
add_sub_cancel'
|
||||
]
|
||||
|
||||
/--
|
||||
Computes the width of a `Skew` rectangle.
|
||||
-/
|
||||
noncomputable def width (r : Skew) : ℝ :=
|
||||
Point.dist r.bl r.br
|
||||
|
||||
/--
|
||||
Computes the height of a `Skew` rectangle.
|
||||
-/
|
||||
noncomputable def height (r : Skew) : ℝ :=
|
||||
Point.dist r.bl r.tl
|
||||
|
||||
/--
|
||||
A mapping from a `Skew` rectangle to the set describing the region enclosed by
|
||||
the rectangle.
|
||||
-/
|
||||
def toSet (r : Skew) : Set Point := sorry
|
||||
|
||||
/-! ## Point -/
|
||||
|
||||
/--
|
||||
A `Point` is a `Skew` rectangle in which all points coincide.
|
||||
-/
|
||||
abbrev AsPoint := Subtype (fun r : Skew => r.tl = r.bl ∧ r.bl = r.br)
|
||||
|
||||
namespace AsPoint
|
||||
|
||||
/--
|
||||
The function mapping a `Skew` rectangle with all points coinciding to a `Point`.
|
||||
-/
|
||||
def toPoint (p : AsPoint) : Point := p.val.tl
|
||||
|
||||
/--
|
||||
The function mapping a `Point` to a `Skew` rectangle with all points coinciding.
|
||||
-/
|
||||
def fromPoint (p : Point) : AsPoint :=
|
||||
have hp : Point.angle p p p = Real.pi / 2 := by
|
||||
unfold Point.angle Real.euclideanAngle
|
||||
simp
|
||||
⟨Skew.mk p p p hp, by simp⟩
|
||||
|
||||
/--
|
||||
An isomorphism between a `Skew` rectangle with all points coinciding and a
|
||||
`Point`.
|
||||
-/
|
||||
def isoPoint : AsPoint ≃ Point :=
|
||||
{
|
||||
toFun := toPoint,
|
||||
invFun := fromPoint,
|
||||
left_inv := by
|
||||
unfold Function.LeftInverse fromPoint toPoint
|
||||
simp only
|
||||
intro p
|
||||
congr
|
||||
· exact p.property.left
|
||||
· rw [p.property.left]
|
||||
exact p.property.right
|
||||
right_inv := by
|
||||
unfold Function.RightInverse Function.LeftInverse fromPoint toPoint
|
||||
simp only
|
||||
intro _
|
||||
triv
|
||||
}
|
||||
|
||||
/--
|
||||
The width of an `AsPoint` is `0`.
|
||||
-/
|
||||
theorem width_eq_zero (p : AsPoint) : p.val.width = 0 := by
|
||||
unfold Skew.width
|
||||
rw [p.property.right]
|
||||
unfold Point.dist
|
||||
simp
|
||||
|
||||
/--
|
||||
The height of an `AsPoint` is `0`.
|
||||
-/
|
||||
theorem height_eq_zero (p : AsPoint) : p.val.height = 0 := by
|
||||
unfold Skew.height
|
||||
rw [p.property.left]
|
||||
unfold Point.dist
|
||||
simp
|
||||
|
||||
end AsPoint
|
||||
|
||||
/-! ## Segment -/
|
||||
|
||||
/--
|
||||
A `Segment` is a `Skew` rectangle in which two of three points coincide.
|
||||
-/
|
||||
abbrev AsSegment := Subtype (fun r : Skew =>
|
||||
(r.tl = r.bl ∧ r.bl ≠ r.br) ∨ (r.tl ≠ r.bl ∧ r.bl = r.br))
|
||||
|
||||
namespace AsSegment
|
||||
|
||||
/--
|
||||
Either the width or height of an `AsSegment` is zero.
|
||||
-/
|
||||
theorem width_or_height_eq_zero (s : AsSegment)
|
||||
: s.val.width = 0 ∨ s.val.height = 0 := by
|
||||
apply Or.elim s.property
|
||||
· intro h
|
||||
refine Or.inr ?_
|
||||
unfold height
|
||||
rw [h.left]
|
||||
unfold Point.dist
|
||||
simp
|
||||
· intro h
|
||||
refine Or.inl ?_
|
||||
unfold width
|
||||
rw [h.right]
|
||||
unfold Point.dist
|
||||
simp
|
||||
|
||||
end AsSegment
|
||||
|
||||
end Skew
|
||||
|
||||
end Geometry.Rectangle
|
||||
|
|
@ -0,0 +1,19 @@
|
|||
import Common.Geometry.Point
|
||||
|
||||
/-! # Common.Geometry.Segment
|
||||
|
||||
A representation of a line segment in the two-dimensional Cartesian plane.
|
||||
-/
|
||||
|
||||
namespace Geometry
|
||||
|
||||
/--
|
||||
A characterization of a line segment.
|
||||
|
||||
The segment comprises of all points between points `p` and `q`, inclusive.
|
||||
-/
|
||||
structure Segment where
|
||||
p : Point
|
||||
q : Point
|
||||
|
||||
end Geometry
|
||||
|
|
@ -0,0 +1,34 @@
|
|||
import Common.Geometry.Point
|
||||
import Common.Set.Partition
|
||||
|
||||
/-! # Common.Geometry.StepFunction
|
||||
|
||||
Characterization of step functions.
|
||||
-/
|
||||
|
||||
namespace Geometry
|
||||
|
||||
open Set Partition
|
||||
|
||||
/--
|
||||
A function `f`, whose domain is a closed interval `[a, b]`, is a `StepFunction`
|
||||
if there exists a `Partition` `P = {x₀, x₁, …, xₙ}` of `[a, b]` such that `f` is
|
||||
constant on each open subinterval of `P`.
|
||||
-/
|
||||
structure StepFunction where
|
||||
p : Partition ℝ
|
||||
toFun : ∀ x ∈ p.toIcc, ℝ
|
||||
const_open_subintervals :
|
||||
∀ (hI : I ∈ p.openSubintervals), ∃ c, ∀ (hy : y ∈ I),
|
||||
toFun y (mem_open_subinterval_mem_closed_interval hI hy) = c
|
||||
|
||||
namespace StepFunction
|
||||
|
||||
/--
|
||||
The ordinate set of the function.
|
||||
-/
|
||||
def toSet (s : StepFunction) : Set Point := sorry
|
||||
|
||||
end StepFunction
|
||||
|
||||
end Geometry
|
||||
|
|
@ -1,4 +1,4 @@
|
|||
import Common.Real.Basic
|
||||
import Common.Real.Geometry
|
||||
import Common.Real.Floor
|
||||
import Common.Real.Rational
|
||||
import Common.Real.Sequence
|
||||
import Common.Real.Trigonometry
|
||||
|
|
@ -1,30 +0,0 @@
|
|||
import Mathlib.Data.Real.Basic
|
||||
|
||||
/-! # Common.Real.Basic
|
||||
|
||||
A collection of basic notational conveniences.
|
||||
-/
|
||||
|
||||
/--
|
||||
An abbreviation of `ℝ²` as the Cartesian product `ℝ × ℝ`.
|
||||
-/
|
||||
notation "ℝ²" => ℝ × ℝ
|
||||
|
||||
namespace Real
|
||||
|
||||
/--
|
||||
Definitionally, the area of a unit circle.
|
||||
|
||||
###### PORT
|
||||
|
||||
As of now, this remains an `axiom`, but it should be replaced by the definition
|
||||
in `Mathlib` once ported to Lean 4.
|
||||
-/
|
||||
axiom pi : ℝ
|
||||
|
||||
/--
|
||||
An abbreviation of `pi` as symbol `π`.
|
||||
-/
|
||||
notation "π" => pi
|
||||
|
||||
end Real
|
||||
|
|
@ -1,3 +0,0 @@
|
|||
import Common.Real.Geometry.Area
|
||||
import Common.Real.Geometry.Basic
|
||||
import Common.Real.Geometry.Rectangle
|
||||
|
|
@ -1,62 +0,0 @@
|
|||
import Mathlib.Data.Real.Sqrt
|
||||
|
||||
import Common.Real.Basic
|
||||
|
||||
/-! # Common.Real.Geometry.Basic
|
||||
|
||||
A collection of useful definitions and theorems around geometry.
|
||||
-/
|
||||
|
||||
namespace Real
|
||||
|
||||
/--
|
||||
The undirected angle at `p₂` between the line segments to `p₁` and `p₃`. If
|
||||
either of those points equals `p₂`, this is `π / 2`.
|
||||
|
||||
###### PORT
|
||||
|
||||
This should be replaced with the original Mathlib `geometry.euclidean.angle`
|
||||
definition once ported.
|
||||
-/
|
||||
axiom angle (p₁ p₂ p₃ : ℝ²) : ℝ
|
||||
|
||||
noncomputable def euclideanAngle (p₁ p₂ p₃ : ℝ²) :=
|
||||
if p₁ = p₂ ∨ p₂ = p₃ then π / 2 else angle p₁ p₂ p₃
|
||||
|
||||
notation "∠" => euclideanAngle
|
||||
|
||||
/--
|
||||
Determine the distance between two points in `ℝ²`.
|
||||
-/
|
||||
noncomputable def dist (x y : ℝ²) :=
|
||||
Real.sqrt ((abs (y.1 - x.1)) ^ 2 + (abs (y.2 - x.2)) ^ 2)
|
||||
|
||||
/--
|
||||
Two sets `S` and `T` are `similar` **iff** there exists a one-to-one
|
||||
correspondence between `S` and `T` such that the distance between any two points
|
||||
`P, Q ∈ S` and corresponding points `P', Q' ∈ T` differ by some constant `α`. In
|
||||
other words, `α|PQ| = |P'Q'|`.
|
||||
-/
|
||||
def similar (S T : Set ℝ²) : Prop :=
|
||||
∃ f : ℝ² → ℝ², Function.Bijective f ∧
|
||||
∃ s : ℝ, ∀ x y : ℝ², x ∈ S ∧ y ∈ T →
|
||||
s * dist x y = dist (f x) (f y)
|
||||
|
||||
/--
|
||||
Two sets are congruent if they are similar with a scaling factor of `1`.
|
||||
-/
|
||||
def congruent (S T : Set (ℝ × ℝ)) : Prop :=
|
||||
∃ f : ℝ² → ℝ², Function.Bijective f ∧
|
||||
∀ x y : ℝ², x ∈ S ∧ y ∈ T →
|
||||
dist x y = dist (f x) (f y)
|
||||
|
||||
/--
|
||||
Any two `congruent` sets must be similar to one another.
|
||||
-/
|
||||
theorem congruent_similar {S T : Set ℝ²} : congruent S T → similar S T := by
|
||||
intro hc
|
||||
let ⟨f, ⟨hf, hs⟩⟩ := hc
|
||||
conv at hs => intro x y hxy; arg 1; rw [← one_mul (dist x y)]
|
||||
exact ⟨f, ⟨hf, ⟨1, hs⟩⟩⟩
|
||||
|
||||
end Real
|
||||
|
|
@ -1,147 +0,0 @@
|
|||
import Common.Real.Geometry.Basic
|
||||
|
||||
/-! # Common.Real.Geometry.Rectangle
|
||||
|
||||
A characterization of a rectangle. This follows the definition as outlined in
|
||||
[^1]. Note that a `Point` and a `LineSegment` are both considered rectangles,
|
||||
with one or both dimensions equal to `0` respectively.
|
||||
|
||||
[^1]: Apostol, Tom M. Calculus, Vol. 1: One-Variable Calculus, with an
|
||||
Introduction to Linear Algebra. 2nd ed. Vol. 1. 2 vols. Wiley, 1991.
|
||||
-/
|
||||
|
||||
namespace Real
|
||||
|
||||
/--
|
||||
A `Rectangle` is characterized by three distinct points and the angle formed
|
||||
between line segments originating from the "bottom left" point.
|
||||
-/
|
||||
structure Rectangle where
|
||||
top_left : ℝ²
|
||||
bottom_left : ℝ²
|
||||
bottom_right : ℝ²
|
||||
forms_right_angle : ∠ top_left bottom_left bottom_right = π / 2
|
||||
|
||||
namespace Rectangle
|
||||
|
||||
/--
|
||||
The top-right corner of the rectangle, oriented with respect to the other
|
||||
vertices.
|
||||
-/
|
||||
def topRight (r : Rectangle) : ℝ² :=
|
||||
( r.top_left.fst + r.bottom_right.fst - r.bottom_left.fst
|
||||
, r.top_left.snd + r.bottom_right.snd - r.bottom_left.snd
|
||||
)
|
||||
|
||||
/--
|
||||
A `Rectangle` is the locus of points bounded by its edges.
|
||||
-/
|
||||
def toSet (r : Rectangle) : Set ℝ² :=
|
||||
sorry
|
||||
|
||||
/--
|
||||
A `Rectangle`'s top side is equal in length to its bottom side.
|
||||
-/
|
||||
theorem dist_top_eq_dist_bottom (r : Rectangle)
|
||||
: dist r.top_left r.topRight = dist r.bottom_left r.bottom_right := by
|
||||
unfold topRight dist
|
||||
repeat rw [add_comm, sub_right_comm, add_sub_cancel']
|
||||
|
||||
/--
|
||||
A `Rectangle`'s left side is equal in length to its right side.
|
||||
-/
|
||||
theorem dist_left_eq_dist_right (r : Rectangle)
|
||||
: dist r.top_left r.bottom_left = dist r.topRight r.bottom_right := by
|
||||
unfold topRight dist
|
||||
repeat rw [
|
||||
sub_sub_eq_add_sub,
|
||||
add_comm,
|
||||
sub_add_eq_sub_sub,
|
||||
sub_right_comm,
|
||||
add_sub_cancel'
|
||||
]
|
||||
|
||||
/--
|
||||
Computes the width of a `Rectangle`.
|
||||
-/
|
||||
noncomputable def width (r : Rectangle) : ℝ :=
|
||||
dist r.bottom_left r.bottom_right
|
||||
|
||||
/--
|
||||
Computes the height of a `Rectangle`.
|
||||
-/
|
||||
noncomputable def height (r : Rectangle) : ℝ :=
|
||||
dist r.bottom_left r.top_left
|
||||
|
||||
end Rectangle
|
||||
|
||||
/--
|
||||
A `Point` is a `Rectangle` in which all points coincide.
|
||||
-/
|
||||
abbrev Point := Subtype (fun r : Rectangle =>
|
||||
r.top_left = r.bottom_left ∧ r.bottom_left = r.bottom_right)
|
||||
|
||||
namespace Point
|
||||
|
||||
/--
|
||||
A `Point` is the set consisting of just itself.
|
||||
-/
|
||||
def toSet (p : Point) : Set ℝ² := p.val.toSet
|
||||
|
||||
/--
|
||||
The width of a `Point` is `0`.
|
||||
-/
|
||||
theorem width_eq_zero (p : Point) : p.val.width = 0 := by
|
||||
unfold Rectangle.width
|
||||
rw [p.property.right]
|
||||
unfold dist
|
||||
simp
|
||||
|
||||
/--
|
||||
The height of a `Point` is `0`.
|
||||
-/
|
||||
theorem height_eq_zero (p : Point) : p.val.height = 0 := by
|
||||
unfold Rectangle.height
|
||||
rw [p.property.left]
|
||||
unfold dist
|
||||
simp
|
||||
|
||||
end Point
|
||||
|
||||
/--
|
||||
A `LineSegment` is a `Rectangle` in which two of the three points coincide.
|
||||
-/
|
||||
abbrev LineSegment := Subtype (fun r : Rectangle =>
|
||||
(r.top_left = r.bottom_left ∧ r.bottom_left ≠ r.bottom_right) ∨
|
||||
(r.top_left ≠ r.bottom_left ∧ r.bottom_left = r.bottom_right))
|
||||
|
||||
namespace LineSegment
|
||||
|
||||
/--
|
||||
A `LineSegment` `s` is the set of points corresponding to the shortest line
|
||||
segment joining the two distinct points of `s`.
|
||||
-/
|
||||
def toSet (s : LineSegment) : Set ℝ² := s.val.toSet
|
||||
|
||||
/--
|
||||
Either the width or height of a `LineSegment` is zero.
|
||||
-/
|
||||
theorem width_or_height_eq_zero (s : LineSegment)
|
||||
: s.val.width = 0 ∨ s.val.height = 0 := by
|
||||
apply Or.elim s.property
|
||||
· intro h
|
||||
refine Or.inr ?_
|
||||
unfold Rectangle.height
|
||||
rw [h.left]
|
||||
unfold dist
|
||||
simp
|
||||
· intro h
|
||||
refine Or.inl ?_
|
||||
unfold Rectangle.width
|
||||
rw [h.right]
|
||||
unfold dist
|
||||
simp
|
||||
|
||||
end LineSegment
|
||||
|
||||
end Real
|
||||
|
|
@ -1,4 +1,4 @@
|
|||
import Common.Real.Basic
|
||||
import Mathlib.Data.Real.Basic
|
||||
|
||||
/-! # Common.Real.Rational
|
||||
|
||||
|
|
|
|||
|
|
@ -0,0 +1,26 @@
|
|||
import Mathlib.Data.Real.Basic
|
||||
|
||||
/-! # Common.Real.Trigonometry
|
||||
|
||||
Additional theorems and definitions useful in the context of trigonometry. Most
|
||||
of these will likely be deleted once the corresponding functions in `Mathlib`
|
||||
are ported to Lean 4.
|
||||
-/
|
||||
|
||||
namespace Real
|
||||
|
||||
/--
|
||||
The standard `π` variable with value `3.14159...`.
|
||||
-/
|
||||
axiom pi : ℝ
|
||||
|
||||
/--
|
||||
The undirected angle at `p₂` between the line segments to `p₁` and `p₃`. If
|
||||
either of those points equals `p₂`, this is `π / 2`.
|
||||
-/
|
||||
axiom angle (p₁ p₂ p₃ : ℝ × ℝ) : ℝ
|
||||
|
||||
noncomputable def euclideanAngle (p₁ p₂ p₃ : ℝ × ℝ) :=
|
||||
if p₁ = p₂ ∨ p₂ = p₃ then pi / 2 else angle p₁ p₂ p₃
|
||||
|
||||
end Real
|
||||
|
|
@ -1,2 +1,2 @@
|
|||
import Common.Set.Basic
|
||||
import Common.Set.Intervals
|
||||
import Common.Set.Partition
|
||||
|
|
@ -1,2 +0,0 @@
|
|||
import Common.Set.Intervals.Partition
|
||||
import Common.Set.Intervals.StepFunction
|
||||
|
|
@ -1,35 +0,0 @@
|
|||
import Common.List.Basic
|
||||
import Common.Set.Intervals.Partition
|
||||
|
||||
/-! # Common.Set.Intervals.StepFunction
|
||||
|
||||
Characterization of step functions.
|
||||
-/
|
||||
|
||||
namespace Set.Intervals
|
||||
|
||||
open Partition
|
||||
|
||||
/--
|
||||
A function `f`, whose domain is a closed interval `[a, b]`, is a `StepFunction`
|
||||
if there exists a `Partition` `P = {x₀, x₁, …, xₙ}` of `[a, b]` such that `f` is
|
||||
constant on each open subinterval of `P`.
|
||||
-/
|
||||
structure StepFunction (α : Type _) [Preorder α] [@DecidableRel α LT.lt] where
|
||||
p : Partition α
|
||||
toFun : ∀ x ∈ p.toIcc, α
|
||||
const_open_subintervals :
|
||||
∀ (hI : I ∈ p.openSubintervals), ∃ c : α, ∀ (hy : y ∈ I),
|
||||
toFun y (mem_open_subinterval_mem_closed_interval hI hy) = c
|
||||
|
||||
namespace StepFunction
|
||||
|
||||
/--
|
||||
The locus of points between the `x`-axis and the function.
|
||||
-/
|
||||
def toSet [Preorder α] [@DecidableRel α LT.lt]
|
||||
(s : StepFunction α) : Set (α × α) := sorry
|
||||
|
||||
end StepFunction
|
||||
|
||||
end Set.Intervals
|
||||
|
|
@ -4,12 +4,12 @@ import Mathlib.Data.Set.Intervals.Basic
|
|||
|
||||
import Common.List.Basic
|
||||
|
||||
/-! # Common.Set.Intervals.Partition
|
||||
/-! # Common.Set.Partition
|
||||
|
||||
Additional theorems and definitions useful in the context of sets.
|
||||
-/
|
||||
|
||||
namespace Set.Intervals
|
||||
namespace Set
|
||||
|
||||
open List
|
||||
|
||||
|
|
@ -134,4 +134,4 @@ theorem mem_open_subinterval_mem_closed_interval
|
|||
|
||||
end Partition
|
||||
|
||||
end Set.Intervals
|
||||
end Set
|
||||
Loading…
Reference in New Issue